1. Combine parsley, olive oil, and 2 tablespoons tarragon in a blender; blend until smooth. Strain mixture through a fine-mesh sieve, pressing on chopped herbs. Reserve 1 tablespoon herb oil (discard herb solids).
2. In a saucepan, combine shallots, broth, and 2 cups water; bring to a boil. Reduce heat to medium low and simmer, partially covered, until shallots are tender (10 minutes). Add peas; simmer until cooked (3 minutes).
3. Working in batches, transfer pea mixture to blender with avocado, yogurt, remaining 2 tablespoons tarragon, and salt and pepper; blend until smooth. (You can also use an immersion blender to blend soup in saucepan.) Divide soup among 4 bowls, drizzle with reserved herb oil, and garnish with additional tarragon leaves.